In this paper, the buckling and vibration behaviour of functionally graded flexoelectric nanobeam is examined. The vibration and buckling formulations of functionally graded nanobeam are developed by using a new theory that’ s presented exclusively for flexoelecteric nano-materials. So by considering Von-Karman strain and forming enthalpy equation based on displacement, polarization and electric potential, electromechanical coupling equations are developed base on Hamilton’ principle. By considering boundary condition of simply support and clamped-clamped and also Euler-Bernoulli beam model, pre-buckling, buckling and the vibration behavior of functionally graded nanobeam affected by flexoelectric will be investigated.

This study presents the effect of porosity on mechanical behaviors of a power distribution functionallygraded beam. The Euler-Bernoulli beam is assumed to describe the kinematic relations and constitutive equations. Because of technical problems, particle size shapes and micro-voids are created during the fabrication which shouldbe taken into consideration. Two porosity models are proposed. The first one describes properties in the explicitform as linear functions of the porosity parameter. The second is a modified model which presents porosity andYoung’ s modulus in an implicit form where the density is assumed as a function of the porosity parameter andYoung’ s modulus as a ratio of mass with porosity to the mass without porosity. The modified proposed model ismore applicable than the first model. The finite element model is developed to solve the problem by using theMATLAB software. Numerical results are presented to show the effects of porosity on mechanical behaviors offunctionally graded beams.

In this paper buckling response of a sandwich (SW) beam containing functionally graded skins and metal (Type-S) or ceramic core (Type-H) is investigated using a third-order zigzag theory. The variation of material properties in functionally graded (FG) layers is quantified through exponential and power laws. The displacements are assumed using higher-order terms along with the zigzag factors to evaluate the effect of shear deformation. In-plane loads are considered. The governing equations are derived using the principle of virtual work. The model achieves stress-free boundaries unlike higher-order shear deformation theories and is C0 continuous so, does not require any post-processing method. The present model shows an accurate variation of transverse stresses in thickness direction due to the inclusion zigzag factor in assumed displacements and is independent of the number of layers in computing the results. Numerical solutions are arrived at by using three noded finite elements with 7DOF/node for sandwich beams. The novelty of the paper lies in presenting a zig-zag buckling analysis for the FGSW beam with thickness stretching. This paper presents the effects of the power law factor, end conditions, aspect ratio, and lamination schemes on the buckling response of FGM sandwich beams. The numerical results are found to be in accordance with the existing results. The buckling strength was improved by increasing the power law factor for Type S beams while the opposite behavior was seen in type H beams for all types of end conditions. The end conditions played a major role in deciding the buckling response of FGSW beams. Exponential law governed FGSW beam exhibited a little higher buckling resistance for Type S beams, while a little lower buckling resistance was found for Type S beams for almost all lamination schemes and end conditions. Some new results are also presented which will serve as a benchmark for future research in a parallel direction.

The buckling and nonlinear free vibration problems of functionally graded porous (FGP) micro-beam resting on an elastic foundation are presented through the nonlocal strain gradient theory (NSGT) and the Euler-Bernoulli beam theory (EBT) with the von-Kármán’s geometrical nonlinearity. The micro-beam is made up of metal and ceramic in which the material properties are assumed to be varied continuously in the thickness direction through a simple exponential law. Two porosity distribution models, including even and uneven distributions, are considered. The governing equation of motion is derived by employing Hamilton’s principle. The analytical expressions of the critical buckling force and nonlinear frequency of the FGP micro-beam with simply supported (S-S) boundary conditions (BCs) are obtained by utilizing the Galerkin technique and the equivalent linearization method (ELM). The reliability of the obtained results has been checked. Effects of the power-law index, the porosity distribution factor, the length-thickness ratio, the material length scale parameter (MLSP), the nonlocal parameter (NP), and the coefficients of the elastic foundation on the buckling and nonlinear free vibration responses of the FGP micro-beam are investigated and discussed in this work.

This study presents an analytical solution for static buckling and free vibration analysis of bi-dimensional functionally graded (2D-FG) metal-ceramic porous beams. To achieve this goal, equations of motion for the beam are derived by using Hamilton's principle and then the derived equations were solved in the framework of Galerkin’s well-known analytical method for solution of equations. The material properties of the beam are variable along with thickness and length according to the power-law function. During the fabrication of functionally graded materials (FGMs), porosities may occur due to technical problems causing micro-voids to appear. Detailed mathematical derivations are presented and numerical investigations are performed, while emphasis is placed on investigating the effect of various parameters such as FG power indexes along both directions of thickness and length, porosity, and slenderness ratios (L/h), on the non-dimensional frequency and static buckling of the beam based on new higher deformation beam theory. The accuracy of the proposed model is validated based on comparisons of the results with the accepted studies. According to the result in both buckling and vibration analysis, the presented modified transverse shear stress along the thickness has shown closer consequences in comparison with TBT.

In the present study, the buckling analysis of single-walled carbon nanotubes (SWCNT) on the basis of a new refined beam theory is analyzed. The SWCNT is modeled as an elastic beam subjected to unidirectional compressive loads. To achieve this aim, the new proposed beam theory has only one unknown variable which leads to one equation similar to Euler beam theory and is also free from any shear correction factors. The equilibrium equation is formulated by the nonlocal elasticity theory in order to predict small-scale effects. The equation is solved by Navier’ s approach by which critical buckling loads are obtained for simple boundary conditions. Finally, to approve the results of the new beam theory, some available well-known references are compared.

In this study, an inverse trigonometric nanobeam theory is applied for the bending, buckling, and free vibration analysis of nanobeams using Eringen’, s nonlocal theory. The present theory satisfies zero shear stress conditions at the top and bottom surfaces of the nanobeam using constitutive relations. Equations of motion are derived by applying Hamilton’, s principle. The present theory is applied for the analysis of functionally graded material nanobeams. All problems are solved by using the Navier technique. For the comparison purpose, the numerical results are generated by using the third shear deformation theory of Reddy, first-order shear deformation theory of Timoshenko, and classical beam theory of Bernoulli-Euler considering the nanosize effects. The present results are found in good agreement with those of higher order theories.

In this study, an efficient finite element model with two degrees of freedom per node is developed for buckling analysis of axially functionally graded (AFG) tapered Timoshenko beams resting on Winkler elastic foundation. For this, the shape functions are exactly acquired through solving the system of equilibrium equations of the Timoshenko beam employing the power series expansions of displacement components. The element stiffness matrix is then formulated by applying the developed shape functions to the total potential energy along the element axis. It is demonstrated that the resulting shape functions, in comparison with Hermitian cubic interpolation functions, are proportional to the mechanical features of the beam element, including the geometrical properties, material characteristics, as well as the critical axial load. An exhaustive numerical example is implemented to clarify the efficiency and simplicity of the proposed mathematical methodology. Furthermore, the effects of end conditions, material gradient, Winkler parameter, tapering ratio, and aspect ratio on the critical buckling load of AFG tapered Timoshenko beam are studied in detail. The numerical outcomes reveal that the elastic foundation enhances the stability characteristics of axially non-homogeneous and homogeneous beams with constant or variable cross-section. Moreover, the results show that the influence of non-uniformity in the cross-section and axially inhomogeneity in material characteristics play significant roles in the linear stability behavior of Timoshenko beams subjected to different boundary conditions.